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doi: 10.3934/jimo.2019125

## Competition in a dual-channel supply chain considering duopolistic retailers with different behaviours

 1 Business School, Beijing Technology and Business University, Beijing 100048, China 2 State Grid Beijing Logistic Supply Company, State Grid Beijing Electric Power Company, Beijing 100054, China 3 School of Management, Capital Normal University, Beijing 100048, China

* Corresponding author: Hongxia Sun

Received  October 2018 Revised  May 2019 Published  October 2019

We study competition in a dual-channel supply chain in which a single supplier sells a single product through its own direct channel and through two different duopolistic retailers. The two retailers have three competitive behaviour patterns: Cournot, Collusion and Stackelberg. Three models are respectively constructed for these patterns, and the optimal decisions for the three patterns are obtained. These optimal solutions are compared, and the effects of certain parameters on the optimal solutions are examined for the three patterns by considering two scenarios: a special case and a general case. In the special case, the equilibrium supply chain structures are analysed, and the optimal quantity and profit are compared for the three different competitive behaviours. Furthermore, both parametric and numerical analyses are presented, and some managerial insights are obtained. We find that in the special case, the Stackelberg game allows the supplier to earn the highest profit, the retailer playing the Collusion game makes the supplier earn the lowest profit, and the Stackelberg leader can gain a first-mover advantage as to the follower. In the general case, the supplier can achieve a higher profit by raising the maximum retail price or holding down the self-price sensitivity factor.

Citation: Hongxia Sun, Yao Wan, Yu Li, Linlin Zhang, Zhen Zhou. Competition in a dual-channel supply chain considering duopolistic retailers with different behaviours. Journal of Industrial & Management Optimization, doi: 10.3934/jimo.2019125
##### References:
 [1] E. Adida and V. DeMiguel, Supply chain competition with multiple manufacturers and retailers, Oper. Res., 59 (2011), 156-172.  doi: 10.1287/opre.1100.0863.  Google Scholar [2] S. Balasubramanian, Mail versus mall: A strategic analysis of competition between direct marketers and conventional retailers, Marketing Science, 17 (1998), 181-195.  doi: 10.1287/mksc.17.3.181.  Google Scholar [3] E. Brynjolfsson, Y. Hu and M. Rahman, Battle of the retail channels: How product selection and geography drive cross-channel competition, Management Science, 55 (2009), 1755-1765.  doi: 10.1287/mnsc.1090.1062.  Google Scholar [4] K. Chen and T. Xiao, Pricing and replenishment policies in a supply chain with competing retailers under different retail behaviors, Computers and Industrial Engineering, 103 (2017), 145-157.  doi: 10.1016/j.cie.2016.11.018.  Google Scholar [5] W. Chiang, D. Chhajed and J. Hess, Direct marketing, indirect profits: A strategic analysis of dual-channel supply-chain design, Management Science, 49 (2003), 1-20.  doi: 10.1287/mnsc.49.1.1.12749.  Google Scholar [6] A. David and E. Adida, Competition and coordination in a two-channel supply chain, Prod. and Oper. Management, 24 (2015), 1358-1370.  doi: 10.1111/poms.12327.  Google Scholar [7] A. Dumrongsiri, M. Fan and A. Jain, et al., A supply chain model with direct and retail channels, European J. Oper. Res., 187 (2008), 691–718. doi: 10.1016/j.ejor.2006.05.044.  Google Scholar [8] X. Han, H. Wu and Q. Yang, et al., Reverse channel selection under remanufacturing risks: Balancing profitability and robustness, International J. of Production Economics, 182 (2016), 63–72. doi: 10.1016/j.ijpe.2016.08.013.  Google Scholar [9] H. Huang, H. Ke and L. Wang, Equilibrium analysis of pricing competition and cooperation in supply chain with one common manufacturer and duopoly retailers, International J. of Production Economics, 178 (2016), 12-21.  doi: 10.1016/j.ijpe.2016.04.022.  Google Scholar [10] C. Ingene and M. Parry, Channel coordination when retailers compete, Marketing Science, 14 (1995), 360-377.  doi: 10.1287/mksc.14.4.360.  Google Scholar [11] M. Lai, H. Yang and E. Cao, et al., Optimal decisions for a dual-channel supply chain under information asymmetry, J. Ind. Manag. Optim., 14 (2018), 1023–1040. doi: 10.3934/jimo.2017088.  Google Scholar [12] N. Modak, S. Panda and S. Sana, Three-echelon supply chain coordination considering duopolistic retailers with perfect quality products, International J. of Production Economics, 182 (2016), 564-578.  doi: 10.1016/j.ijpe.2015.05.021.  Google Scholar [13] I. Moon and X. Feng, Supply chain coordination with a single supplier and multiple retailers considering customer arrival times and route selection, Transportation Research Part E: Logistics and Transportation Review, 106 (2017), 78-97.  doi: 10.1016/j.tre.2017.08.004.  Google Scholar [14] R. Sadeghi, A. Taleizadeh and F. Chan, et al., Coordinating and pricing decisions in two competitive reverse supply chains with different channel structures, International J. of Production Research, 57 (2019), 2601–2625. doi: 10.1080/00207543.2018.1551637.  Google Scholar [15] A. Sadigh, S. Chaharsooghi and M. Sheikhmohammady, A game theoretic approach to coordination of pricing, advertising, and inventory decisions in a competitive supply chain, J. Ind. Manag. Optim., 12 (2016), 337-355. doi: 10.3934/jimo.2016.12.337.  Google Scholar [16] A. Vinhas and E. Anderson, How potential conflict drives channel structure: Concurrent (direct and indirect) channels, Journal of Marketing Research, 42 (2005), 507-515.  doi: 10.1509/jmkr.2005.42.4.507.  Google Scholar [17] L. Wang, H. Song and D. Zhang, et al., Pricing decisions for complementary products in a fuzzy dual-channel supply chain, J. Ind. Manag. Optim., 15 (2019), 343–364. doi: 10.3934/jimo.2018046.  Google Scholar [18] S. Wang, J. Wang and Y. Zhou, Channel coordination with different competitive duopolistic retail behaviour and non-linear demand function, International J. of Manag. Science and Engineering Manag., 7 (2012), 119-127.  doi: 10.1080/17509653.2012.10671214.  Google Scholar [19] S. Wang, Y. Zhou and J. Min, et al., Coordination of cooperative advertising models in a one-manufacturer two-retailer supply chain system, Computers and Industrial Engineering, 61 (2011), 1053–1071. doi: 10.1016/j.cie.2011.06.020.  Google Scholar [20] W. Wang, G. Li and T. Cheng, Channel selection in a supply chain with a multi-channel retailer: The role of channel operating costs, International J. of Production Economics, 173 (2016), 54-65.  doi: 10.1016/j.ijpe.2015.12.004.  Google Scholar [21] W. Wang, P. Zhang and J. Ding, et al., Closed-loop supply chain network equilibrium model with retailer-collection under legislation, J. Ind. Manag. Optim., 15 (2019), 199–219. doi: 10.3934/jimo.2018039.  Google Scholar [22] D. Wu, B. Zhang and O. Baron, A trade credit model with asymmetric competing retailers, Production and Oper. Manag., 28 (2019), 206-231.  doi: 10.1111/poms.12882.  Google Scholar [23] T. Xiao, T. Choi and T. Cheng, Product variety and channel structure strategy for a retailer-Stackelberg supply chain, European J. Oper. Res., 233 (2014), 114-124.  doi: 10.1016/j.ejor.2013.08.038.  Google Scholar [24] G. Xu, B. Dan and X. Zhang, et al., Coordinating a dual-channel supply chain with riskaverse under a two-way revenue sharing contract, International J. of Production Economics, 147 (2014), 171–179. doi: 10.1016/j.ijpe.2013.09.012.  Google Scholar [25] L. Yang, J. Ji and K. Chen, Advertising games on national brand and store brand in a dual-channel supply chain, J. Ind. Manag. Optim., 14 (2018), 105-134.  doi: 10.3934/jimo.2017039.  Google Scholar [26] S. Yang and Y. Zhou, Two-echelon supply chain models: Considering duopolistic retailers' different competitive behaviors, International J. of Production Economics, 103 (2006), 104-116.  doi: 10.1016/j.ijpe.2005.06.001.  Google Scholar [27] Z. Yao, S. Leung and K. Lai, Manufacturer's revenue-sharing contract and retail competition, European J. Oper. Res., 186 (2008), 637-651.  doi: 10.1016/j.ejor.2007.01.049.  Google Scholar

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##### References:
 [1] E. Adida and V. DeMiguel, Supply chain competition with multiple manufacturers and retailers, Oper. Res., 59 (2011), 156-172.  doi: 10.1287/opre.1100.0863.  Google Scholar [2] S. Balasubramanian, Mail versus mall: A strategic analysis of competition between direct marketers and conventional retailers, Marketing Science, 17 (1998), 181-195.  doi: 10.1287/mksc.17.3.181.  Google Scholar [3] E. Brynjolfsson, Y. Hu and M. Rahman, Battle of the retail channels: How product selection and geography drive cross-channel competition, Management Science, 55 (2009), 1755-1765.  doi: 10.1287/mnsc.1090.1062.  Google Scholar [4] K. Chen and T. Xiao, Pricing and replenishment policies in a supply chain with competing retailers under different retail behaviors, Computers and Industrial Engineering, 103 (2017), 145-157.  doi: 10.1016/j.cie.2016.11.018.  Google Scholar [5] W. Chiang, D. Chhajed and J. Hess, Direct marketing, indirect profits: A strategic analysis of dual-channel supply-chain design, Management Science, 49 (2003), 1-20.  doi: 10.1287/mnsc.49.1.1.12749.  Google Scholar [6] A. David and E. Adida, Competition and coordination in a two-channel supply chain, Prod. and Oper. Management, 24 (2015), 1358-1370.  doi: 10.1111/poms.12327.  Google Scholar [7] A. Dumrongsiri, M. Fan and A. Jain, et al., A supply chain model with direct and retail channels, European J. Oper. Res., 187 (2008), 691–718. doi: 10.1016/j.ejor.2006.05.044.  Google Scholar [8] X. Han, H. Wu and Q. Yang, et al., Reverse channel selection under remanufacturing risks: Balancing profitability and robustness, International J. of Production Economics, 182 (2016), 63–72. doi: 10.1016/j.ijpe.2016.08.013.  Google Scholar [9] H. Huang, H. Ke and L. Wang, Equilibrium analysis of pricing competition and cooperation in supply chain with one common manufacturer and duopoly retailers, International J. of Production Economics, 178 (2016), 12-21.  doi: 10.1016/j.ijpe.2016.04.022.  Google Scholar [10] C. Ingene and M. Parry, Channel coordination when retailers compete, Marketing Science, 14 (1995), 360-377.  doi: 10.1287/mksc.14.4.360.  Google Scholar [11] M. Lai, H. Yang and E. Cao, et al., Optimal decisions for a dual-channel supply chain under information asymmetry, J. Ind. Manag. Optim., 14 (2018), 1023–1040. doi: 10.3934/jimo.2017088.  Google Scholar [12] N. Modak, S. Panda and S. Sana, Three-echelon supply chain coordination considering duopolistic retailers with perfect quality products, International J. of Production Economics, 182 (2016), 564-578.  doi: 10.1016/j.ijpe.2015.05.021.  Google Scholar [13] I. Moon and X. Feng, Supply chain coordination with a single supplier and multiple retailers considering customer arrival times and route selection, Transportation Research Part E: Logistics and Transportation Review, 106 (2017), 78-97.  doi: 10.1016/j.tre.2017.08.004.  Google Scholar [14] R. Sadeghi, A. Taleizadeh and F. Chan, et al., Coordinating and pricing decisions in two competitive reverse supply chains with different channel structures, International J. of Production Research, 57 (2019), 2601–2625. doi: 10.1080/00207543.2018.1551637.  Google Scholar [15] A. Sadigh, S. Chaharsooghi and M. Sheikhmohammady, A game theoretic approach to coordination of pricing, advertising, and inventory decisions in a competitive supply chain, J. Ind. Manag. Optim., 12 (2016), 337-355. doi: 10.3934/jimo.2016.12.337.  Google Scholar [16] A. Vinhas and E. Anderson, How potential conflict drives channel structure: Concurrent (direct and indirect) channels, Journal of Marketing Research, 42 (2005), 507-515.  doi: 10.1509/jmkr.2005.42.4.507.  Google Scholar [17] L. Wang, H. Song and D. Zhang, et al., Pricing decisions for complementary products in a fuzzy dual-channel supply chain, J. Ind. Manag. Optim., 15 (2019), 343–364. doi: 10.3934/jimo.2018046.  Google Scholar [18] S. Wang, J. Wang and Y. Zhou, Channel coordination with different competitive duopolistic retail behaviour and non-linear demand function, International J. of Manag. Science and Engineering Manag., 7 (2012), 119-127.  doi: 10.1080/17509653.2012.10671214.  Google Scholar [19] S. Wang, Y. Zhou and J. Min, et al., Coordination of cooperative advertising models in a one-manufacturer two-retailer supply chain system, Computers and Industrial Engineering, 61 (2011), 1053–1071. doi: 10.1016/j.cie.2011.06.020.  Google Scholar [20] W. Wang, G. Li and T. Cheng, Channel selection in a supply chain with a multi-channel retailer: The role of channel operating costs, International J. of Production Economics, 173 (2016), 54-65.  doi: 10.1016/j.ijpe.2015.12.004.  Google Scholar [21] W. Wang, P. Zhang and J. Ding, et al., Closed-loop supply chain network equilibrium model with retailer-collection under legislation, J. Ind. Manag. Optim., 15 (2019), 199–219. doi: 10.3934/jimo.2018039.  Google Scholar [22] D. Wu, B. Zhang and O. Baron, A trade credit model with asymmetric competing retailers, Production and Oper. Manag., 28 (2019), 206-231.  doi: 10.1111/poms.12882.  Google Scholar [23] T. Xiao, T. Choi and T. Cheng, Product variety and channel structure strategy for a retailer-Stackelberg supply chain, European J. Oper. Res., 233 (2014), 114-124.  doi: 10.1016/j.ejor.2013.08.038.  Google Scholar [24] G. Xu, B. Dan and X. Zhang, et al., Coordinating a dual-channel supply chain with riskaverse under a two-way revenue sharing contract, International J. of Production Economics, 147 (2014), 171–179. doi: 10.1016/j.ijpe.2013.09.012.  Google Scholar [25] L. Yang, J. Ji and K. Chen, Advertising games on national brand and store brand in a dual-channel supply chain, J. Ind. Manag. Optim., 14 (2018), 105-134.  doi: 10.3934/jimo.2017039.  Google Scholar [26] S. Yang and Y. Zhou, Two-echelon supply chain models: Considering duopolistic retailers' different competitive behaviors, International J. of Production Economics, 103 (2006), 104-116.  doi: 10.1016/j.ijpe.2005.06.001.  Google Scholar [27] Z. Yao, S. Leung and K. Lai, Manufacturer's revenue-sharing contract and retail competition, European J. Oper. Res., 186 (2008), 637-651.  doi: 10.1016/j.ejor.2007.01.049.  Google Scholar
Supply chain structure
Effect of $a_k$ on $\Pi_0$ and ($\Pi_1$-$\Pi_2$) in three patterns
Effect of $\theta_k$ on $q_0$ and $\Pi_0$ in three patterns
Effect of $\theta_k$ on $q_1$ and $\Pi_1$ in three patterns
Effect of $\theta_k$ on $q_2$ and $\Pi_2$ in three patterns
Effect of $\theta_k$ or $\beta$ on ($\Pi_1$-$\Pi_2$) in three patterns
Effect of $\beta$ on $q_k$ or $\Pi_k$ in three patterns
The equilibrium supply chain structure about the three different competitive behaviors
 Structure Wholesale supplier Dual-channel Monopoly retailer Cournot $\delta\leq\frac{2\beta}{2\theta+\beta}$ $\frac{2\beta}{2\theta+\beta}<\delta<\frac{\theta_0}{\beta}$ $\delta\geq\frac{\theta_0}{\beta}$ $q^{co}_0$ - $\frac{2\beta(a-c)-a_0(2\theta+\beta)}{2[2\beta^2-\theta_0(2\theta+\beta)]}$ $\frac{a_0}{2\theta_0}$ $q^{co}_1$ $\frac{a-c}{2(2\theta+\beta)}$ $\frac{V}{2[2\beta^2-\theta_0(2\theta+\beta)]}$ - $q^{co}_2$ $\frac{a-c}{2(2\theta+\beta)}$ $\frac{V}{2[2\beta^2-\theta_0(2\theta+\beta)]}$ - $w^{co}$ $\frac{a+c}{2}$ $\frac{a+c}{2}$ - $p_0^{co}$ - $\frac{a_0}{2}$ $\frac{a_0}{2}$ $p_1^{co}$ $\frac{(a+c)(\theta+\beta)+2a\theta}{2(2\theta+\beta)}$ $\frac{a+c}{2}+\frac{V\theta}{2[2\beta^2-\theta_0(2\theta+\beta)]}$ - $p_2^{co}$ $\frac{(a+c)(\theta+\beta)+2a\theta}{2(2\theta+\beta)}$ $\frac{a+c}{2}+\frac{V\theta}{2[2\beta^2-\theta_0(2\theta+\beta)]}$ - $\Pi_0^{co}$ $\frac{(a-c)^2}{2(2\theta+\beta)}$ $\frac{2(a-c)(a_0\beta+V)-a_0^2(2\theta+\beta)}{4[2\beta^2-\theta_0(2\theta+\beta)]}$ $\frac{a_0^2}{4\theta_0}$ $\Pi_{1}^{co}$ $\frac{\theta(a-c)^2}{4(2\theta+\beta)^2}$ $\frac{\theta V^2}{4[2\beta^2-\theta_0(2\theta+\beta)]^2}$ - $\Pi_{2}^{co}$ $\frac{\theta(a-c)^2}{4(2\theta+\beta)^2}$ $\frac{\theta V^2}{4[2\beta^2-\theta_0(2\theta+\beta)]^2}$ - Collusion $\delta\leq\frac{\beta}{\theta+\beta}$ $\frac{\beta}{\theta+\beta}<\delta<\frac{\theta_0}{\beta}$ $\delta\geq\frac{\theta_0}{\beta}$ $q^{cn}_0$ - $\frac{a_0(\theta+\beta)-\beta(a-c)}{2[\theta_0(\theta+\beta)-\beta^2]}$ $\frac{a_0}{2\theta_0}$ $q^{cn}_1$ $\frac{a-c}{4(\theta+\beta)}$ $\frac{V}{4[\beta^2-\theta_0(\theta+\beta)]}$ - $q^{cn}_2$ $\frac{a-c}{4(\theta+\beta)}$ $\frac{V}{4[\beta^2-\theta_0(\theta+\beta)]}$ - $w^{cn}$ $\frac{a+c}{2}$ $\frac{a+c}{2}$ - $p_0^{cn}$ - $\frac{a_0}{2}$ $\frac{a_0}{2}$ $p_1^{cn}$ $\frac{3a+c}{4}$ $\frac{a+c}{2}+\frac{(\theta+\beta)V}{4[\beta^2-\theta_0(\theta+\beta)]}$ - $p_2^{cn}$ $\frac{3a+c}{4}$ $\frac{a+c}{2}+\frac{(\theta+\beta)V}{4[\beta^2-\theta_0(\theta+\beta)]}$ - $\Pi_0^{cn}$ $\frac{(a-c)^2}{4(\theta+\beta)}$ $\frac{(a-c)(V+a_0\beta)-a_0^2(\theta+\beta)}{4[\beta^2-\theta_0(\theta+\beta)]}$ $\frac{a_0^2}{4\theta_0}$ $\Pi_{1}^{cn}$ $\frac{(a-c)^2}{16(\theta+\beta)}$ $\frac{V^2(\theta+\beta)}{16[\beta^2-\theta_0(\theta+\beta)]^2}$ - $\Pi_{2}^{cn}$ $\frac{(a-c)^2}{16(\theta+\beta)}$ $\frac{V^2(\theta+\beta)}{16[\beta^2-\theta_0(\theta+\beta)]^2}$ - Stackelberg $\delta\leq\frac{\beta U}{T}$ $\frac{\beta U}{T}<\delta<\frac{\theta_0}{\beta}$ $\delta\geq\frac{\theta_0}{\beta}$ $q^{st}_0$ - $\frac{\beta U(a-c)-a_0T}{2(\beta^2 U-\theta_0T)}$ $\frac{a_0}{2\theta_0}$ $q^{st}_1$ $\frac{\theta(a-c)(2\theta-\beta)}{T}$ $\frac{\theta V(2\theta-\beta)}{\beta^2 U-\theta_0T}$ - $q^{st}_2$ $\frac{(a-c)(4\theta^2-\beta^2-2\theta\beta)}{2T}$ $\frac{V(4\theta^2-\beta^2-2\theta\beta)}{2(\beta^2 U-\theta_0T)}$ - $w^{st}$ $\frac{a+c}{2}$ $\frac{a+c}{2}$ - $p_0^{st}$ - $\frac{a_0}{2}$ $\frac{a_0}{2}$ $p_1^{st}$ $a-\frac{(a-c)(4\theta^3+2\theta^2\beta-\beta^3-2\theta\beta^2)}{2T}$ $\frac{a+c}{2}+\frac{V(2\theta^2-\beta^2)(2\theta-\beta)}{2(\beta^2 U-\theta_0T)}$ - $p_2^{st}$ $a-\frac{\theta(a-c)(4\theta^2+2\theta\beta-3\beta^2)}{2T}$ $\frac{a+c}{2}+\frac{\theta V(4\theta^2-\beta^2-2\theta\beta)}{2(\beta^2 U-\theta_0T)}$ - $\Pi_0^{st}$ $\frac{U(a-c)^2}{4T}$ $\frac{U(a-c)(a_0\beta+V)-a_0^2T}{4(\beta^2 U-\theta_0T)}$ $\frac{a_0^2}{4\theta_0}$ $\Pi_{1}^{st}$ $\frac{\theta(a-c)^2(2\theta-\beta)(4\theta^3-2\theta^2\beta+\beta^3-2\theta\beta^2)}{2T^2}$ $\frac{T(2\theta-\beta)^2V^2}{8(\beta^2 U-\theta_0T)^2}$ - $\Pi_{2}^{st}$ $\frac{\theta(a-c)^2(4\theta^2-\beta^2-2\theta\beta)^2}{4T^2}$ $\frac{\theta V^2(4\theta^2-2\theta\beta-\beta^2)^2}{4(\beta^2 U-\theta_0T)^2}$ - where $T = 4\theta(2\theta^2-\beta^2)$, $U = 8\theta^2-4\theta\beta-\beta^2$, $V = a_0\beta-\theta_0(a-c)$
 Structure Wholesale supplier Dual-channel Monopoly retailer Cournot $\delta\leq\frac{2\beta}{2\theta+\beta}$ $\frac{2\beta}{2\theta+\beta}<\delta<\frac{\theta_0}{\beta}$ $\delta\geq\frac{\theta_0}{\beta}$ $q^{co}_0$ - $\frac{2\beta(a-c)-a_0(2\theta+\beta)}{2[2\beta^2-\theta_0(2\theta+\beta)]}$ $\frac{a_0}{2\theta_0}$ $q^{co}_1$ $\frac{a-c}{2(2\theta+\beta)}$ $\frac{V}{2[2\beta^2-\theta_0(2\theta+\beta)]}$ - $q^{co}_2$ $\frac{a-c}{2(2\theta+\beta)}$ $\frac{V}{2[2\beta^2-\theta_0(2\theta+\beta)]}$ - $w^{co}$ $\frac{a+c}{2}$ $\frac{a+c}{2}$ - $p_0^{co}$ - $\frac{a_0}{2}$ $\frac{a_0}{2}$ $p_1^{co}$ $\frac{(a+c)(\theta+\beta)+2a\theta}{2(2\theta+\beta)}$ $\frac{a+c}{2}+\frac{V\theta}{2[2\beta^2-\theta_0(2\theta+\beta)]}$ - $p_2^{co}$ $\frac{(a+c)(\theta+\beta)+2a\theta}{2(2\theta+\beta)}$ $\frac{a+c}{2}+\frac{V\theta}{2[2\beta^2-\theta_0(2\theta+\beta)]}$ - $\Pi_0^{co}$ $\frac{(a-c)^2}{2(2\theta+\beta)}$ $\frac{2(a-c)(a_0\beta+V)-a_0^2(2\theta+\beta)}{4[2\beta^2-\theta_0(2\theta+\beta)]}$ $\frac{a_0^2}{4\theta_0}$ $\Pi_{1}^{co}$ $\frac{\theta(a-c)^2}{4(2\theta+\beta)^2}$ $\frac{\theta V^2}{4[2\beta^2-\theta_0(2\theta+\beta)]^2}$ - $\Pi_{2}^{co}$ $\frac{\theta(a-c)^2}{4(2\theta+\beta)^2}$ $\frac{\theta V^2}{4[2\beta^2-\theta_0(2\theta+\beta)]^2}$ - Collusion $\delta\leq\frac{\beta}{\theta+\beta}$ $\frac{\beta}{\theta+\beta}<\delta<\frac{\theta_0}{\beta}$ $\delta\geq\frac{\theta_0}{\beta}$ $q^{cn}_0$ - $\frac{a_0(\theta+\beta)-\beta(a-c)}{2[\theta_0(\theta+\beta)-\beta^2]}$ $\frac{a_0}{2\theta_0}$ $q^{cn}_1$ $\frac{a-c}{4(\theta+\beta)}$ $\frac{V}{4[\beta^2-\theta_0(\theta+\beta)]}$ - $q^{cn}_2$ $\frac{a-c}{4(\theta+\beta)}$ $\frac{V}{4[\beta^2-\theta_0(\theta+\beta)]}$ - $w^{cn}$ $\frac{a+c}{2}$ $\frac{a+c}{2}$ - $p_0^{cn}$ - $\frac{a_0}{2}$ $\frac{a_0}{2}$ $p_1^{cn}$ $\frac{3a+c}{4}$ $\frac{a+c}{2}+\frac{(\theta+\beta)V}{4[\beta^2-\theta_0(\theta+\beta)]}$ - $p_2^{cn}$ $\frac{3a+c}{4}$ $\frac{a+c}{2}+\frac{(\theta+\beta)V}{4[\beta^2-\theta_0(\theta+\beta)]}$ - $\Pi_0^{cn}$ $\frac{(a-c)^2}{4(\theta+\beta)}$ $\frac{(a-c)(V+a_0\beta)-a_0^2(\theta+\beta)}{4[\beta^2-\theta_0(\theta+\beta)]}$ $\frac{a_0^2}{4\theta_0}$ $\Pi_{1}^{cn}$ $\frac{(a-c)^2}{16(\theta+\beta)}$ $\frac{V^2(\theta+\beta)}{16[\beta^2-\theta_0(\theta+\beta)]^2}$ - $\Pi_{2}^{cn}$ $\frac{(a-c)^2}{16(\theta+\beta)}$ $\frac{V^2(\theta+\beta)}{16[\beta^2-\theta_0(\theta+\beta)]^2}$ - Stackelberg $\delta\leq\frac{\beta U}{T}$ $\frac{\beta U}{T}<\delta<\frac{\theta_0}{\beta}$ $\delta\geq\frac{\theta_0}{\beta}$ $q^{st}_0$ - $\frac{\beta U(a-c)-a_0T}{2(\beta^2 U-\theta_0T)}$ $\frac{a_0}{2\theta_0}$ $q^{st}_1$ $\frac{\theta(a-c)(2\theta-\beta)}{T}$ $\frac{\theta V(2\theta-\beta)}{\beta^2 U-\theta_0T}$ - $q^{st}_2$ $\frac{(a-c)(4\theta^2-\beta^2-2\theta\beta)}{2T}$ $\frac{V(4\theta^2-\beta^2-2\theta\beta)}{2(\beta^2 U-\theta_0T)}$ - $w^{st}$ $\frac{a+c}{2}$ $\frac{a+c}{2}$ - $p_0^{st}$ - $\frac{a_0}{2}$ $\frac{a_0}{2}$ $p_1^{st}$ $a-\frac{(a-c)(4\theta^3+2\theta^2\beta-\beta^3-2\theta\beta^2)}{2T}$ $\frac{a+c}{2}+\frac{V(2\theta^2-\beta^2)(2\theta-\beta)}{2(\beta^2 U-\theta_0T)}$ - $p_2^{st}$ $a-\frac{\theta(a-c)(4\theta^2+2\theta\beta-3\beta^2)}{2T}$ $\frac{a+c}{2}+\frac{\theta V(4\theta^2-\beta^2-2\theta\beta)}{2(\beta^2 U-\theta_0T)}$ - $\Pi_0^{st}$ $\frac{U(a-c)^2}{4T}$ $\frac{U(a-c)(a_0\beta+V)-a_0^2T}{4(\beta^2 U-\theta_0T)}$ $\frac{a_0^2}{4\theta_0}$ $\Pi_{1}^{st}$ $\frac{\theta(a-c)^2(2\theta-\beta)(4\theta^3-2\theta^2\beta+\beta^3-2\theta\beta^2)}{2T^2}$ $\frac{T(2\theta-\beta)^2V^2}{8(\beta^2 U-\theta_0T)^2}$ - $\Pi_{2}^{st}$ $\frac{\theta(a-c)^2(4\theta^2-\beta^2-2\theta\beta)^2}{4T^2}$ $\frac{\theta V^2(4\theta^2-2\theta\beta-\beta^2)^2}{4(\beta^2 U-\theta_0T)^2}$ - where $T = 4\theta(2\theta^2-\beta^2)$, $U = 8\theta^2-4\theta\beta-\beta^2$, $V = a_0\beta-\theta_0(a-c)$
Partial derivatives of optimal quantity with respect to $a_k$ in three patterns
 Cournot $a_0$ $a_1$ $a_2$ $q^{co}_0$ $\frac{-E}{2P}$ $\frac{\beta(2\theta_2-\beta)}{2P}$ $\frac{\beta(2\theta_1-\beta)}{2P}$ $q^{co}_1$ $\frac{-\beta(\beta-2\theta_2)}{2P}$ $\frac{P(2\theta_2D+E)-\beta^2(2\theta_2-\beta)^2D}{2PDE}$ $\frac{-P(\beta D+E)-\beta^2(2\theta_1-\beta)(2\theta_2-\beta)D}{2PDE}$ $q^{co}_2$ $\frac{-\beta(\beta-2\theta_1)}{2P}$ $\frac{-P(\beta D+E)-\beta^2(2\theta_1-\beta)(2\theta_2-\beta)D}{2PDE}$ $\frac{P(2\theta_1D+E)-\beta^2(2\theta_1-\beta)^2D}{2PDE}$ Collusion $a_0$ $a_1$ $a_2$ $q^{cn}_0$ $\frac{\theta_1\theta_2-\beta^2}{Q}$ $\frac{-\beta(\theta_2-\beta)}{2Q}$ $\frac{-\beta(\theta_1-\beta)}{2Q}$ $q^{cn}_1$ $\frac{\beta(\beta-\theta_2)}{2Q}$ $\frac{Q(2\theta_2G+H)+2G\beta^2(\theta_2-\beta)^2}{4QGH}$ $\frac{-Q(2\beta G+H)+2G\beta^2(\theta_1-\beta)(\theta_2-\beta)}{4QGH}$ $q^{cn}_2$ $\frac{\beta(\beta-\theta_1)}{2Q}$ $\frac{-Q(2\beta G+H)+2G\beta^2(\theta_1-\beta)(\theta_2-\beta)}{4QGH}$ $\frac{Q(2\theta_1G+H)+2G\beta^2(\theta_2-\beta)^2}{4QGH}$ Stackelberg $a_0$ $a_1$ $a_2$ $q^{st}_0$ $\frac{2M\theta_2}{R}$ $\frac{-\theta_2\beta(2\theta_2-\beta)}{R}$ $\frac{-\beta S}{2R}$ $q^{st}_1$ $\frac{\theta_2\beta(\beta-2\theta_2)}{R}$ $\frac{R(\theta_2 N+2\theta_2M)+\theta_2\beta^2(2\theta_2-\beta)^2N}{2RMN}$ $\frac{-R(4M\theta_2+\beta N)+\beta^2(2\theta_2-\beta)SN}{4RMN}$ $q^{st}_2$ $\frac{\beta(2\theta_2\beta-E)}{2R}$ $\frac{-R(4M\theta_2+\beta N)+\beta^2(2\theta_2-\beta)SN}{4RMN}$ $\frac{RNE+S^2(\beta^2N-2M\theta_0\theta_2)}{4\theta_2RMN}$ where $P=\beta^2D-\theta_0E$, $Q=\theta_0H-\beta^2G$, $R=4M\theta_0\theta_2-\beta^2N$, $S=E-2\theta_2\beta$.
 Cournot $a_0$ $a_1$ $a_2$ $q^{co}_0$ $\frac{-E}{2P}$ $\frac{\beta(2\theta_2-\beta)}{2P}$ $\frac{\beta(2\theta_1-\beta)}{2P}$ $q^{co}_1$ $\frac{-\beta(\beta-2\theta_2)}{2P}$ $\frac{P(2\theta_2D+E)-\beta^2(2\theta_2-\beta)^2D}{2PDE}$ $\frac{-P(\beta D+E)-\beta^2(2\theta_1-\beta)(2\theta_2-\beta)D}{2PDE}$ $q^{co}_2$ $\frac{-\beta(\beta-2\theta_1)}{2P}$ $\frac{-P(\beta D+E)-\beta^2(2\theta_1-\beta)(2\theta_2-\beta)D}{2PDE}$ $\frac{P(2\theta_1D+E)-\beta^2(2\theta_1-\beta)^2D}{2PDE}$ Collusion $a_0$ $a_1$ $a_2$ $q^{cn}_0$ $\frac{\theta_1\theta_2-\beta^2}{Q}$ $\frac{-\beta(\theta_2-\beta)}{2Q}$ $\frac{-\beta(\theta_1-\beta)}{2Q}$ $q^{cn}_1$ $\frac{\beta(\beta-\theta_2)}{2Q}$ $\frac{Q(2\theta_2G+H)+2G\beta^2(\theta_2-\beta)^2}{4QGH}$ $\frac{-Q(2\beta G+H)+2G\beta^2(\theta_1-\beta)(\theta_2-\beta)}{4QGH}$ $q^{cn}_2$ $\frac{\beta(\beta-\theta_1)}{2Q}$ $\frac{-Q(2\beta G+H)+2G\beta^2(\theta_1-\beta)(\theta_2-\beta)}{4QGH}$ $\frac{Q(2\theta_1G+H)+2G\beta^2(\theta_2-\beta)^2}{4QGH}$ Stackelberg $a_0$ $a_1$ $a_2$ $q^{st}_0$ $\frac{2M\theta_2}{R}$ $\frac{-\theta_2\beta(2\theta_2-\beta)}{R}$ $\frac{-\beta S}{2R}$ $q^{st}_1$ $\frac{\theta_2\beta(\beta-2\theta_2)}{R}$ $\frac{R(\theta_2 N+2\theta_2M)+\theta_2\beta^2(2\theta_2-\beta)^2N}{2RMN}$ $\frac{-R(4M\theta_2+\beta N)+\beta^2(2\theta_2-\beta)SN}{4RMN}$ $q^{st}_2$ $\frac{\beta(2\theta_2\beta-E)}{2R}$ $\frac{-R(4M\theta_2+\beta N)+\beta^2(2\theta_2-\beta)SN}{4RMN}$ $\frac{RNE+S^2(\beta^2N-2M\theta_0\theta_2)}{4\theta_2RMN}$ where $P=\beta^2D-\theta_0E$, $Q=\theta_0H-\beta^2G$, $R=4M\theta_0\theta_2-\beta^2N$, $S=E-2\theta_2\beta$.
The optimal solutions for three different competitive behaviors
 Optimal $q_0$ $q_1$ $q_2$ $w$ $p_0$ $p_1$ $p_2$ $\Pi_0$ $\Pi_1$ $\Pi_2$ Cournot $4.83$ $3.44$ $2.90$ $14.25$ $8.00$ $17.69$ $20.06$ $116.36$ $11.85$ $16.86$ Collusion $5.29$ $2.81$ $2.60$ $14.13$ $8.00$ $18.24$ $20.74$ $108.01$ $11.57$ $17.23$ Stackelberg $4.76$ $3.58$ $2.90$ $14.23$ $8.00$ $17.59$ $20.03$ $117.35$ $12.04$ $16.82$
 Optimal $q_0$ $q_1$ $q_2$ $w$ $p_0$ $p_1$ $p_2$ $\Pi_0$ $\Pi_1$ $\Pi_2$ Cournot $4.83$ $3.44$ $2.90$ $14.25$ $8.00$ $17.69$ $20.06$ $116.36$ $11.85$ $16.86$ Collusion $5.29$ $2.81$ $2.60$ $14.13$ $8.00$ $18.24$ $20.74$ $108.01$ $11.57$ $17.23$ Stackelberg $4.76$ $3.58$ $2.90$ $14.23$ $8.00$ $17.59$ $20.03$ $117.35$ $12.04$ $16.82$
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